Polar Coordinates and Rose Curves

This video tutorial explains how to determine the area of a region using double integrals in polar coordinates. It covers the process of replacing differential elements in polar form and provides a detailed example of calculating the area of a rose petal. The tutorial demonstrates using a graphing calculator to find the interval for Theta and walks through the integration process, including the use of power-reducing formulas. The video concludes with a summary and a preview of the next example.

What is the additional factor introduced in polar coordinates when determining area using double integrals?

• r dr dθ

• dydx

• r dθ

• dxdy

In the context of the rose curve, what is the expression for R?

• 2 cos 3θ

• 2 sin 3θ

• 3 sin 2θ

• 3 cos 2θ

What is the starting value of R when tracing out one petal of the rose curve?

• 1

• 2

• 0

• 3

What mode should the graphing calculator be in to determine the interval for theta?

• Rectangular mode

• Cartesian mode

• Polar mode

• Radian mode

What is the interval for theta that traces out one petal of the rose curve?

• 0 to π/3

• 0 to π/2

• 0 to 2π

• 0 to π

What is the period of the basic sine function?

• π/2

• 2π

• π

• 3π

How do you determine the interval for theta by hand for the rose curve?

• Set 3θ = π and solve for θ

• Set θ = π and solve for 3θ

• Set 3θ = 2π and solve for θ

• Set θ = 2π and solve for 3θ

What formula is used to simplify sin²θ during integration?

• Power-reducing formula

• Double angle formula

• Sum-to-product formula

• Product-to-sum formula

What is the antiderivative of cos(6θ) in terms of θ?

• sin(6θ)/6

• -cos(6θ)/6

• cos(6θ)/6

• -sin(6θ)/6

What is the area of one petal of the rose curve?

• π/2

• π/3

• 2π

• π